# Properties

 Label 1600.2.a.u Level $1600$ Weight $2$ Character orbit 1600.a Self dual yes Analytic conductor $12.776$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,2,Mod(1,1600)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$12.7760643234$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 40) Fricke sign: $$+1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + 2 q^{3} - 2 q^{7} + q^{9}+O(q^{10})$$ q + 2 * q^3 - 2 * q^7 + q^9 $$q + 2 q^{3} - 2 q^{7} + q^{9} - 4 q^{11} - 4 q^{13} - 4 q^{19} - 4 q^{21} + 2 q^{23} - 4 q^{27} - 2 q^{29} - 8 q^{33} - 4 q^{37} - 8 q^{39} + 2 q^{41} - 6 q^{43} + 6 q^{47} - 3 q^{49} + 4 q^{53} - 8 q^{57} - 12 q^{59} + 10 q^{61} - 2 q^{63} + 14 q^{67} + 4 q^{69} - 8 q^{71} + 8 q^{73} + 8 q^{77} - 16 q^{79} - 11 q^{81} + 2 q^{83} - 4 q^{87} + 6 q^{89} + 8 q^{91} + 16 q^{97} - 4 q^{99}+O(q^{100})$$ q + 2 * q^3 - 2 * q^7 + q^9 - 4 * q^11 - 4 * q^13 - 4 * q^19 - 4 * q^21 + 2 * q^23 - 4 * q^27 - 2 * q^29 - 8 * q^33 - 4 * q^37 - 8 * q^39 + 2 * q^41 - 6 * q^43 + 6 * q^47 - 3 * q^49 + 4 * q^53 - 8 * q^57 - 12 * q^59 + 10 * q^61 - 2 * q^63 + 14 * q^67 + 4 * q^69 - 8 * q^71 + 8 * q^73 + 8 * q^77 - 16 * q^79 - 11 * q^81 + 2 * q^83 - 4 * q^87 + 6 * q^89 + 8 * q^91 + 16 * q^97 - 4 * q^99

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 2.00000 0 0 0 −2.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.a.u 1
4.b odd 2 1 1600.2.a.f 1
5.b even 2 1 1600.2.a.d 1
5.c odd 4 2 320.2.c.b 2
8.b even 2 1 400.2.a.b 1
8.d odd 2 1 200.2.a.d 1
15.e even 4 2 2880.2.f.i 2
20.d odd 2 1 1600.2.a.v 1
20.e even 4 2 320.2.c.c 2
24.f even 2 1 1800.2.a.s 1
24.h odd 2 1 3600.2.a.k 1
40.e odd 2 1 200.2.a.b 1
40.f even 2 1 400.2.a.g 1
40.i odd 4 2 80.2.c.a 2
40.k even 4 2 40.2.c.a 2
56.e even 2 1 9800.2.a.d 1
60.l odd 4 2 2880.2.f.h 2
80.i odd 4 2 1280.2.f.b 2
80.j even 4 2 1280.2.f.a 2
80.s even 4 2 1280.2.f.f 2
80.t odd 4 2 1280.2.f.e 2
120.i odd 2 1 3600.2.a.bb 1
120.m even 2 1 1800.2.a.j 1
120.q odd 4 2 360.2.f.c 2
120.w even 4 2 720.2.f.e 2
280.n even 2 1 9800.2.a.bf 1
280.y odd 4 2 1960.2.g.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.c.a 2 40.k even 4 2
80.2.c.a 2 40.i odd 4 2
200.2.a.b 1 40.e odd 2 1
200.2.a.d 1 8.d odd 2 1
320.2.c.b 2 5.c odd 4 2
320.2.c.c 2 20.e even 4 2
360.2.f.c 2 120.q odd 4 2
400.2.a.b 1 8.b even 2 1
400.2.a.g 1 40.f even 2 1
720.2.f.e 2 120.w even 4 2
1280.2.f.a 2 80.j even 4 2
1280.2.f.b 2 80.i odd 4 2
1280.2.f.e 2 80.t odd 4 2
1280.2.f.f 2 80.s even 4 2
1600.2.a.d 1 5.b even 2 1
1600.2.a.f 1 4.b odd 2 1
1600.2.a.u 1 1.a even 1 1 trivial
1600.2.a.v 1 20.d odd 2 1
1800.2.a.j 1 120.m even 2 1
1800.2.a.s 1 24.f even 2 1
1960.2.g.b 2 280.y odd 4 2
2880.2.f.h 2 60.l odd 4 2
2880.2.f.i 2 15.e even 4 2
3600.2.a.k 1 24.h odd 2 1
3600.2.a.bb 1 120.i odd 2 1
9800.2.a.d 1 56.e even 2 1
9800.2.a.bf 1 280.n even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1600))$$:

 $$T_{3} - 2$$ T3 - 2 $$T_{7} + 2$$ T7 + 2 $$T_{11} + 4$$ T11 + 4 $$T_{13} + 4$$ T13 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 2$$
$5$ $$T$$
$7$ $$T + 2$$
$11$ $$T + 4$$
$13$ $$T + 4$$
$17$ $$T$$
$19$ $$T + 4$$
$23$ $$T - 2$$
$29$ $$T + 2$$
$31$ $$T$$
$37$ $$T + 4$$
$41$ $$T - 2$$
$43$ $$T + 6$$
$47$ $$T - 6$$
$53$ $$T - 4$$
$59$ $$T + 12$$
$61$ $$T - 10$$
$67$ $$T - 14$$
$71$ $$T + 8$$
$73$ $$T - 8$$
$79$ $$T + 16$$
$83$ $$T - 2$$
$89$ $$T - 6$$
$97$ $$T - 16$$